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Rafael O. Ruggiero

Publications and source records attributed to Rafael O. Ruggiero.

8 recordsLinked to original sources
arXiv2026

An upper bound for the mean Lorentz force of magnetic flows without conjugate points in terms of the geodesic curvature of horocycles

We study magnetic flows on closed Riemannian surfaces of genus at least two. Let $(M,g)$ be a surface without focal points and let $Ω$ be a magnetic field. We prove that, above the Mañé critical value, the absence of conjugate points imposes geometric restrictions on the magnetic flow. First, we obtain a bound for the integral of the Lorentz force in terms of the geodesic curvature of the horocycles of the underlying Riemannian metric. As a consequence, a lower bound on the Gaussian curvature yields an explicit bound in terms of the area of the surface. The proofs combine the global geometry of magnetic geodesics with Liouville's formula for the geodesic curvature in the orthogonal coordinates given by the level sets and geodesics of a Busemann function on the universal cover.

math.DG↗
arXiv2026

On rigidity of Finsler manifolds without conjugate points and with constant $S$-curvature

We study rigidity phenomena in closed Finsler manifolds without conjugate points under assumptions on the $S$-curvature. We prove that a closed $C^ω$ Finsler manifold with constant $S$-curvature, continuous Green bundles, and admitting a hyperbolic closed geodesic must be Riemannian. In the $C^\infty$ setting, the same conclusion holds under the additional assumption that the geodesic flow is transitive. As a consequence, we obtain rigidity results for Finsler manifolds with uniform visibility universal covering. Our approach is based on the analysis of the Cartan vector field as a Jacobi field and its interaction with the geometry of Green bundles.

math.DG↗
arXiv2024

Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

We consider here the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is either the action of shift $T=σ$ on the symbolic space $M=\{1,2,...,d\}^\mathbb{N}$, or, $T$ describes the action of a $d$ to $1$ expanding transformation $T:S^1 \to S^1$ of class $C^{1+α}$ (\,for example $x \to T(x) =d\, x $ (mod $1) $\,), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibrium probabilities for Hölder potentials $A:M \to \mathbb{R}$ is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When $T=σ$ and $M=\{0,1\}^\mathbb{N}$ such basis exists. In a different direction, we also consider the KL-divergence $D_{KL}(μ_1,μ_2)$ for a pair of equilibrium probabilities. If $D_{KL}(μ_1,μ_2)=0$, then $μ_1=μ_2$. Although $D_{KL}$ is not a metric in $\mathcal{N}$, it describes the proximity between $μ_1$ and $μ_2$. A natural problem is: for a fixed probability $μ_1\in \mathcal{N}$ consider the probability $μ_2$ in a convex set of probabilities in $\mathcal{N}$ which minimizes $D_{KL}(μ_1,μ_2)$. This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in $\mathcal{N}$, a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.

math.DS↗
arXiv2023

The sectional curvature of the infinite dimensional manifold of Hölder equilibrium prababilities

Here we consider the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is the shift and $M=\{1,2,...,d\}^\mathbb{N}$. It is known that the infinite-dimensional manifold $\mathcal{N}$ of Hölder equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a normalized Hölder potential $A$ denote by $μ_A \in \mathcal{N}$ the associated equilibrium probability. The set of tangent vectors $X$ to the manifold $\mathcal{N}$ at the point $μ_A$ coincides with the kernel of the Ruelle operator for $A$. The Riemannian norm $|X|=|X|_A$ of the vector $X$, which is tangent to $\mathcal{N}$ at the point $μ_A$, is described via the asymptotic variance, that is, satisfies $|X|^2\,\,= \langle X, X \rangle =\lim_{n \to \infty} \frac{1}{n} \int (\sum_{i=0}^{n-1} X\circ T^i )^2 \,d μ_A$. Consider an orthonormal basis $X_i$, $i \in \mathbb{N}$, for the tangent space at $μ_A$. Given two unit tangent vectors $X$ and $Y$ the curvature $K(X,Y)$ satisfies $\,\,\,\,K(X,Y) = \frac{1}{4}[\, \sum_{i=1}^\infty ( \int X \,Y\, X_i \,d μ_A)^2 - \sum_{i=1}^\infty \int X^2 X_i \,d μ_A\, \,\int Y^2 X_i \,d μ_A \,].$ When the equilibrium probabilities $μ_A$ is the set of invariant Markov probabilities on $\{0,1\}^\mathbb{N}\subset \mathcal{N}$, introducing an orthonormal basis $\hat{a}_y$, indexed by finite words $y$, we show explicit expressions for $K(\hat{a}_x,\hat{a}_z)$, which is a finite sum. These values can be positive or negative depending on $A$ and the words $x$ and $z$. Words $x,z$ with large length can eventually produce large negative curvature $K(\hat{a}_x,\hat{a}_z)$. If $x, z$ do not begin with the same letter, then $K(\hat{a}_x,\hat{a}_z)=0$.

math.DS↗
arXiv2020

Geodesic flows modeled by expansive flows: Compact surfaces without conjugate points and continuous Green bundles

We study the geodesic flow of a compact surface without conjugate points and genus greater than one and continuous Green bundles. Identifying each strip of bi-asymptotic geodesics induces an equivalence relation on the unit tangent bundle. Its quotient space is shown to carry the structure of a 3-dimensional compact manifold. This manifold carries a canonically defined continuous flow which is expansive, time-preserving semi-conjugate to the geodesic flow, and has a local product structure. An essential step towards the proof of these properties is to study regularity properties of the horospherical foliations and to show that they are indeed tangent to the Green subbundles. As an application it is shown that the geodesic flow has a unique measure of maximal entropy.

math.DS↗
arXiv2017

Geodesic flows modeled by expansive flows

Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time-parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.

math.DS↗
arXiv2010

Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics

We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_λ(p,v)=\frac{1}{2}g_{p}(v,v)- λω_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $ω$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-ω_{p}(v)$ has support in a unique closed geodesic $γ$; and the curvature is negative at every point of $M$ but at the points of $γ$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large deviation function is of polynomial type, the power of the polynomial function depends on the way the curvature goes to zero in a neighborhood of $γ$. This results has interesting counterparts in one-dimensional dynamics with indifferent fixed points and convex billiards with flat points in the boundary of the billiard. A previous estimate by N. Anantharaman of the large deviation function in terms of the Peierl's barrier of the Aubry-Mather measure is crucial for our result.

math.DS↗